Fitting : Voting and the Hough Transform Monday, Feb 14 Prof. Kristen Grauman UT-Austin

Now: Fitting Want to associate a model with observed features [Fig from Marszalek & Schmid, 2007] For example, the model could be a line, a circle, or an arbitrary shape. Kristen Grauman

Fitting: Main idea Choose a parametric model to represent a set of features Membership criterion is not local Can’t tell whether a point belongs to a given model just by looking at that point Three main questions: What model represents this set of features best? Which of several model instances gets which feature? How many model instances are there? Computational complexity is important It is infeasible to examine every possible set of parameters and every possible combination of features Slide credit: L. Lazebnik

Example: Line fitting Why fit lines? Many objects characterized by presence of straight lines Wait, why aren’t we done just by running edge detection? Kristen Grauman

Extra edge points (clutter), multiple models: –which points go with which line, if any? Only some parts of each line detected, and some parts are missing: –how to find a line that bridges missing evidence? Noise in measured edge points, orientations: –how to detect true underlying parameters? Difficulty of line fitting Kristen Grauman

Voting It’s not feasible to check all combinations of features by fitting a model to each possible subset. Voting is a general technique where we let the features vote for all models that are compatible with it. –Cycle through features, cast votes for model parameters. –Look for model parameters that receive a lot of votes. Noise & clutter features will cast votes too, but typically their votes should be inconsistent with the majority of “good” features. Kristen Grauman

Fitting lines: Hough transform Given points that belong to a line, what is the line? How many lines are there? Which points belong to which lines? Hough Transform is a voting technique that can be used to answer all of these questions. Main idea: 1. Record vote for each possible line on which each edge point lies. 2. Look for lines that get many votes. Kristen Grauman

Finding lines in an image: Hough space Connection between image (x,y) and Hough (m,b) spaces A line in the image corresponds to a point in Hough space To go from image space to Hough space: –given a set of points (x,y), find all (m,b) such that y = mx + b x y m b m0m0 b0b0 image spaceHough (parameter) space Slide credit: Steve Seitz

Finding lines in an image: Hough space Connection between image (x,y) and Hough (m,b) spaces A line in the image corresponds to a point in Hough space To go from image space to Hough space: –given a set of points (x,y), find all (m,b) such that y = mx + b What does a point (x 0, y 0 ) in the image space map to? x y m b image spaceHough (parameter) space –Answer: the solutions of b = -x 0 m + y 0 –this is a line in Hough space x0x0 y0y0 Slide credit: Steve Seitz

Finding lines in an image: Hough space What are the line parameters for the line that contains both (x 0, y 0 ) and (x 1, y 1 )? It is the intersection of the lines b = –x 0 m + y 0 and b = –x 1 m + y 1 x y m b image spaceHough (parameter) space x0x0 y0y0 b = –x 1 m + y 1 (x 0, y 0 ) (x 1, y 1 )

Finding lines in an image: Hough algorithm How can we use this to find the most likely parameters (m,b) for the most prominent line in the image space? Let each edge point in image space vote for a set of possible parameters in Hough space Accumulate votes in discrete set of bins; parameters with the most votes indicate line in image space. x y m b image spaceHough (parameter) space

Polar representation for lines : perpendicular distance from line to origin : angle the perpendicular makes with the x-axis Point in image space  sinusoid segment in Hough space [0,0] Issues with usual (m,b) parameter space: can take on infinite values, undefined for vertical lines. Image columns Image rows Kristen Grauman

Hough line demo a/hough.htmlhttp:// a/hough.html

Hough transform algorithm Using the polar parameterization: Basic Hough transform algorithm 1.Initialize H[d,  ]=0 2.for each edge point I[x,y] in the image for  = [  min to  max ] // some quantization H[d,  ] += 1 3.Find the value(s) of (d,  ) where H[d,  ] is maximum 4.The detected line in the image is given by H: accumulator array (votes) d  Time complexity (in terms of number of votes per pt)? Source: Steve Seitz

Original imageCanny edges Vote space and top peaks Kristen Grauman

Showing longest segments found Kristen Grauman

Impact of noise on Hough Image space edge coordinates Votes  x y d What difficulty does this present for an implementation?

Image space edge coordinates Votes Impact of noise on Hough Here, everything appears to be “noise”, or random edge points, but we still see peaks in the vote space.

Extensions Extension 1: Use the image gradient 1.same 2.for each edge point I[x,y] in the image  = gradient at (x,y) H[d,  ] += 1 3.same 4.same (Reduces degrees of freedom) Extension 2 give more votes for stronger edges Extension 3 change the sampling of (d,  ) to give more/less resolution Extension 4 The same procedure can be used with circles, squares, or any other shape

Extensions Extension 1: Use the image gradient 1.same 2.for each edge point I[x,y] in the image compute unique (d,  ) based on image gradient at (x,y) H[d,  ] += 1 3.same 4.same (Reduces degrees of freedom) Extension 2 give more votes for stronger edges (use magnitude of gradient) Extension 3 change the sampling of (d,  ) to give more/less resolution Extension 4 The same procedure can be used with circles, squares, or any other shape… Source: Steve Seitz

Hough transform for circles For a fixed radius r, unknown gradient direction Circle: center (a,b) and radius r Image space Hough space Kristen Grauman

Hough transform for circles For a fixed radius r, unknown gradient direction Circle: center (a,b) and radius r Image space Hough space Intersection: most votes for center occur here. Kristen Grauman

Hough transform for circles For an unknown radius r, unknown gradient direction Circle: center (a,b) and radius r Hough space Image space b a r ? Kristen Grauman

Hough transform for circles For an unknown radius r, unknown gradient direction Circle: center (a,b) and radius r Hough space Image space b a r Kristen Grauman

Hough transform for circles For an unknown radius r, known gradient direction Circle: center (a,b) and radius r Hough space Image space θ x Kristen Grauman

Hough transform for circles For every edge pixel (x,y) : For each possible radius value r: For each possible gradient direction θ: // or use estimated gradient at (x,y) a = x – r cos(θ) // column b = y + r sin(θ) // row H[a,b,r] += 1 end Check out online demo : Time complexity per edgel? Kristen Grauman

Original Edges Example: detecting circles with Hough Votes: Penny Note: a different Hough transform (with separate accumulators) was used for each circle radius (quarters vs. penny).

Original Edges Example: detecting circles with Hough Votes: Quarter Combined detections Coin finding sample images from: Vivek Kwatra

Example: iris detection Hemerson Pistori and Eduardo Rocha Costa Gradient+thresholdHough space (fixed radius) Max detections Kristen Grauman

Example: iris detection An Iris Detection Method Using the Hough Transform and Its Evaluation for Facial and Eye Movement, by Hideki Kashima, Hitoshi Hongo, Kunihito Kato, Kazuhiko Yamamoto, ACCV Kristen Grauman

Voting: practical tips Minimize irrelevant tokens first Choose a good grid / discretization Vote for neighbors, also (smoothing in accumulator array) Use direction of edge to reduce parameters by 1 To read back which points voted for “winning” peaks, keep tags on the votes. Too coarse Too fine ? Kristen Grauman

Hough transform: pros and cons Pros All points are processed independently, so can cope with occlusion, gaps Some robustness to noise: noise points unlikely to contribute consistently to any single bin Can detect multiple instances of a model in a single pass Cons Complexity of search time increases exponentially with the number of model parameters Non-target shapes can produce spurious peaks in parameter space Quantization: can be tricky to pick a good grid size Kristen Grauman

Generalized Hough Transform Model image Vote space Novel image x x x x x Now suppose those colors encode gradient directions… What if we want to detect arbitrary shapes? Intuition: Ref. point Displacement vectors Kristen Grauman

Define a model shape by its boundary points and a reference point. [Dana H. Ballard, Generalizing the Hough Transform to Detect Arbitrary Shapes, 1980] x a p1p1 θ p2p2 θ At each boundary point, compute displacement vector: r = a – p i. Store these vectors in a table indexed by gradient orientation θ. Generalized Hough Transform Offline procedure: Model shape θ θ … … … Kristen Grauman

p1p1 θ θ For each edge point: Use its gradient orientation θ to index into stored table Use retrieved r vectors to vote for reference point Generalized Hough Transform Detection procedure: Assuming translation is the only transformation here, i.e., orientation and scale are fixed. x θ θ Novel image θ θ … … … θ x x xx Kristen Grauman

Generalized Hough for object detection Instead of indexing displacements by gradient orientation, index by matched local patterns. B. Leibe, A. Leonardis, and B. Schiele, Combined Object Categorization and Segmentation with an Implicit Shape Model, ECCV Workshop on Statistical Learning in Computer Vision 2004Combined Object Categorization and Segmentation with an Implicit Shape Model training image “visual codeword” with displacement vectors Source: L. Lazebnik

Instead of indexing displacements by gradient orientation, index by “visual codeword” B. Leibe, A. Leonardis, and B. Schiele, Combined Object Categorization and Segmentation with an Implicit Shape Model, ECCV Workshop on Statistical Learning in Computer Vision 2004Combined Object Categorization and Segmentation with an Implicit Shape Model test image Source: L. Lazebnik Generalized Hough for object detection

Summary Grouping/segmentation useful to make a compact representation and merge similar features –associate features based on defined similarity measure and clustering objective Fitting problems require finding any supporting evidence for a model, even within clutter and missing features. –associate features with an explicit model Voting approaches, such as the Hough transform, make it possible to find likely model parameters without searching all combinations of features. –Hough transform approach for lines, circles, …, arbitrary shapes defined by a set of boundary points, recognition from patches. Kristen Grauman